precal question

Question

Simplify using the Binomial Theorem.

(x + h)5 − x5

h

 

Answer 1

For the expression (x + h)⁵ - x⁵ we can see the x⁵ terms will cancel out, but we must expand the bracketed part first to see what remains.

As I've already worked out a table of coefficients in my Pascal's Triangle from my last answer up to power 10, I know that for a power of 5 the coefficients are: 1-----5--------10---------10--------5-----1

Hence I can expand (x + h)⁵ to get x⁵ + 5x⁴h + 10x³h² + 10x²h³ + 5xh⁴ + h⁵

And therefore (x + h)⁵ - x⁵ = 5x⁴h + 10x³h² + 10x²h³ + 5xh⁴ + h⁵

Which is equal to h(5x⁴ + 10x³h + 10x²h² + 5xh³ + h⁴)